Thus the motivation for the pair truncation of the SM.For example, the (generalized) seniority scheme [1]}, [2]}, [3]}, [4]}, [5]} and broken pair model [6]}, [7]}, [8]}, which work very well for nearly spherical nuclei.
The dominant building blocks are nucleon pairs coupled to angular momentum zero (denoted by \(S\)  pairs), due to the strong monopole pairing interaction.
The ground state of a semimagic even-even nucleus can be described by an \(S\)  pair condensate, and the low-lying excited states are interpreted as breaking of the \(S\)  pairs.
In the interacting boson model [9]}, [10]}, [11]}, [12]}, the spin-zero \(S\)  pair and spin-two (\(D\) ) pair are mapped to \(s\)  and \(d\)  bosons, key ingredients in collective states of the vibrational, rotational, and \(\gamma \) -soft nuclei.
A fully fermionic treatment is found
in the nucleon-pair approximation (NPA) of the SM [13]}, [14]}, built by nucleon pairs with spins zero, two, four (denoted by \(G\) ), six (denoted by \(I\) ), etc.
NPA calculations successfully reproduce low-lying states in nuclei from spherical, via transitional, and finally to deformed regions [15]}, [16]}, and the important role played by the \(G\)  (and sometimes \(I\) ) pair in well-deformed nuclei has been demonstrated [17]}, [18]}.
An important ingredient for deformed and transitional nuclides was the extraction of good pairs from Hartree-Fock states [16]}, [17]}.
Although NPA configuration spaces are much smaller than full SM ones, and despite recent significant speed-ups in
codes driven largely by going from a \(J\) -coupled scheme to an \(M\)  or \(J_z\) -scheme [21]}, [22]},
the computation with high-spin \(G\)  and \(I\)  pairs is still too burdensome for most of deformed nuclei across the nuclear chart.
